The hard thresholding operator in (REF ) discards a large important part of a vector when the vector is dense.
This means some important information of the vector has been lost in the process of hard thresholding.
As pointed out in [1]}, [2]}, the hard thresholding should be applied to a compressible vector instead of a dense vector in order to avoid a big loss of information.
Note that the vector \(x^p - \lambda \nabla f(x^p)\)  in (REF ) is usually dense since the search direction \(-\nabla f(x^p)\)  is not necessarily sparse.
This motivates us to adopt the partial gradient instead of the full gradient as a search direction in order to generate the following sparse or compressible vector:
\( u^{p} := x^p - \lambda {\cal H}_q (\nabla f(x^p)) \) 
